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<?xml version="1.0" encoding="utf-8"?>
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<?xml version="1.0" encoding="utf-8"?>
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<?xml version="1.0" encoding="utf-8"?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-04-01 mer. 16:14 -->
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<!-- 2020-04-05 dim. 19:43 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Compensating the gravity forces to start at steady state</title>
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@ -270,9 +269,9 @@ In order to start the simulation at steady state in presence of gravity:
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We don’t inject any perturbations and no reference tracking.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeReferences();
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeController();
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<pre class="src src-matlab"> initializeReferences();
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeController();
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</pre>
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</div>
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@ -280,8 +279,8 @@ initializeController();
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We include the gravity and log all the signals to display.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
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<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
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</pre>
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</div>
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</div>
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@ -296,28 +295,28 @@ Let’s simulate the system without any compensation of gravity forces.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod();
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initializeSample();
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<pre class="src src-matlab"> initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod();
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initializeSample();
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
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<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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sim_no_compensation = simout;
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<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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sim_no_compensation = simout;
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</pre>
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</div>
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@ -342,16 +341,16 @@ We here wish to simulate the system in order to compute the required force in ea
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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<pre class="src src-matlab"> initializeGround();
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initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
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</pre>
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</div>
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@ -359,13 +358,13 @@ initializeSample(<span class="org-string">'type'</span>, <span class="org-string
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We simulate for a short time period (all the bodies are solid, so nothing should move).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.1'</span>);
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<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.1'</span>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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</pre>
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</div>
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@ -373,7 +372,7 @@ We simulate for a short time period (all the bodies are solid, so nothing should
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Verification that nothing is moving by looking at the maximum displacement of the sample:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">max(max(simout.Em.En.Data))
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<pre class="src src-matlab"> max(max(simout.Em.En.Data))
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</pre>
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</div>
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@ -386,7 +385,7 @@ Verification that nothing is moving by looking at the maximum displacement of th
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We here show the measured total force/torque applied at the location of each joint.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">data2orgtable([Fgm 0 0 0; Ftym; Fym; Fsm], {<span class="org-string">'Granite'</span>, <span class="org-string">'Translation Stage'</span>, <span class="org-string">'Tilt Stage'</span>, <span class="org-string">'Sample'</span>}, {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>}, <span class="org-string">' %.1e '</span>);
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<pre class="src src-matlab"> data2orgtable([Fgm 0 0 0; Ftym; Fym; Fsm], {<span class="org-string">'Granite'</span>, <span class="org-string">'Translation Stage'</span>, <span class="org-string">'Tilt Stage'</span>, <span class="org-string">'Sample'</span>}, {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>}, <span class="org-string">' %.1e '</span>);
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</pre>
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</div>
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@ -463,7 +462,7 @@ We here show the measured total force/torque applied at the location of each joi
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</table>
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<div class="org-src-container">
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<pre class="src src-matlab">data2orgtable([Fhm; Fnm], {<span class="org-string">'Micro-Hexapod'</span>, <span class="org-string">'Nano-Hexapod'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}, <span class="org-string">' %.1e '</span>);
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<pre class="src src-matlab"> data2orgtable([Fhm; Fnm], {<span class="org-string">'Micro-Hexapod'</span>, <span class="org-string">'Nano-Hexapod'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}, <span class="org-string">' %.1e '</span>);
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</pre>
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</div>
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@ -523,7 +522,7 @@ We here show the measured total force/torque applied at the location of each joi
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We save these forces in <code>Foffset.mat</code>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">save(<span class="org-string">'mat/Foffset.mat'</span>, <span class="org-string">'Fgm'</span>, <span class="org-string">'Ftym'</span>, <span class="org-string">'Fym'</span>, <span class="org-string">'Fzm'</span>, <span class="org-string">'Fhm'</span>, <span class="org-string">'Fnm'</span>, <span class="org-string">'Fsm'</span>);
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<pre class="src src-matlab"> save(<span class="org-string">'mat/Foffset.mat'</span>, <span class="org-string">'Fgm'</span>, <span class="org-string">'Ftym'</span>, <span class="org-string">'Fym'</span>, <span class="org-string">'Fzm'</span>, <span class="org-string">'Fhm'</span>, <span class="org-string">'Fnm'</span>, <span class="org-string">'Fsm'</span>);
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</pre>
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</div>
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</div>
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@ -537,16 +536,16 @@ We save these forces in <code>Foffset.mat</code>.
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We now initialize the stages with the option <code>Foffset</code>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeTy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeRy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeRz(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeMicroHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeSample(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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<pre class="src src-matlab"> initializeGround();
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initializeGranite(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeTy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeRy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeRz(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeMicroHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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initializeSample(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
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</pre>
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</div>
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@ -554,14 +553,14 @@ initializeSample(<span class="org-string">'Foffset'</span>, <span class="org-con
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And we simulate the system for 0.5 seconds.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
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<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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sim_compensation = simout;
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<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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sim_compensation = simout;
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</pre>
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</div>
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@ -579,16 +578,19 @@ Verification that nothing is moving
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<div id="outline-container-orgb714922" class="outline-2">
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<h2 id="orgb714922"><span class="section-number-2">5</span> Conclusion</h2>
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<div class="outline-text-2" id="text-5">
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<div class="important">
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<p>
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This initialization technique permits to compute the required forces/torques to be applied in each joint in order to compensate for gravity forces.
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This initialization should be redone for each configuration (change of sample mass, change of tilt angle), but not when changing the stiffness of joints, for instant when changing from lorentz based nano-hexapod or piezo based.
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</p>
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</div>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-01 mer. 16:14</p>
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<p class="date">Created: 2020-04-05 dim. 19:43</p>
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</div>
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</body>
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</html>
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@ -4,7 +4,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-04-03 ven. 17:55 -->
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<!-- 2020-04-05 dim. 19:43 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Determination of the optimal nano-hexapod’s stiffness</title>
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@ -249,7 +249,7 @@
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<ul>
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<li><a href="#org84aa9ce">1. Spindle Rotation Speed</a>
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<ul>
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<li><a href="#org0a41acf">1.1. Initialization</a></li>
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<li><a href="#org37d23dd">1.1. Initialization</a></li>
|
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<li><a href="#org75fea48">1.2. Identification when rotating at maximum speed</a></li>
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<li><a href="#orgf19758e">1.3. Change of dynamics</a></li>
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</ul>
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@ -264,7 +264,7 @@
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</li>
|
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<li><a href="#orgd6382d8">3. Payload “Impedance” Effect</a>
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<ul>
|
||||
<li><a href="#org37d23dd">3.1. Initialization</a></li>
|
||||
<li><a href="#orgc1416f3">3.1. Initialization</a></li>
|
||||
<li><a href="#orgc8cd70d">3.2. Identification of the dynamics while change the payload dynamics</a></li>
|
||||
<li><a href="#org4da36db">3.3. Change of dynamics for the primary controller</a>
|
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<ul>
|
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@ -321,21 +321,21 @@ The rotation speed will have an effect due to the Coriolis effect.
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||||
</p>
|
||||
</div>
|
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|
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<div id="outline-container-org0a41acf" class="outline-3">
|
||||
<h3 id="org0a41acf"><span class="section-number-3">1.1</span> Initialization</h3>
|
||||
<div id="outline-container-org37d23dd" class="outline-3">
|
||||
<h3 id="org37d23dd"><span class="section-number-3">1.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
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||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
<pre class="src src-matlab"> initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -343,7 +343,7 @@ initializeMirror();
|
||||
We use a sample mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
<pre class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
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@ -352,10 +352,10 @@ We don’t include disturbances in this model as it adds complexity to the s
|
||||
We however include gravity.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeController();
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeController();
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -368,7 +368,7 @@ initializeController();
|
||||
We identify the dynamics for the following spindle rotation speeds <code>Rz_rpm</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Rz_rpm = linspace(0, 60, 6);
|
||||
<pre class="src src-matlab"> Rz_rpm = linspace(0, 60, 6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -376,7 +376,7 @@ We identify the dynamics for the following spindle rotation speeds <code>Rz_rpm<
|
||||
And for the following nano-hexapod actuator stiffness <code>Ks</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
<pre class="src src-matlab"> Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -389,38 +389,45 @@ And for the following nano-hexapod actuator stiffness <code>Ks</code>:
|
||||
We plot the change of dynamics due to the change of the spindle rotation speed (from 0rpm to 60rpm):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org2b01fbe">1</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#org9bfe588">2</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#org9f887c8">3</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org5926aca">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org2b01fbe">2</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#org9bfe588">3</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#org9f887c8">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org5926aca">5</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org3c83afc" class="figure">
|
||||
<p><img src="figs/opti_stiffness_iff_root_locus.png" alt="opti_stiffness_iff_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Root Locus plot for IFF control when not rotating (in red) and when rotating at 60rpm (in blue) for 4 different nano-hexapod stiffnesses (<a href="./figs/opti_stiffness_iff_root_locus.png">png</a>, <a href="./figs/opti_stiffness_iff_root_locus.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org2b01fbe" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_iff.png" alt="opt_stiffness_wz_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_iff.png">png</a>, <a href="./figs/opt_stiffness_wz_iff.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_iff.png">png</a>, <a href="./figs/opt_stiffness_wz_iff.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9bfe588" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_dvf.png" alt="opt_stiffness_wz_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_dvf.png">png</a>, <a href="./figs/opt_stiffness_wz_dvf.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_dvf.png">png</a>, <a href="./figs/opt_stiffness_wz_dvf.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9f887c8" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_fx_dx.png" alt="opt_stiffness_wz_fx_dx.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_wz_fx_dx.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_wz_fx_dx.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5926aca" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_coupling.png" alt="opt_stiffness_wz_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Change of Coupling from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_y\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_coupling.png">png</a>, <a href="./figs/opt_stiffness_wz_coupling.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Change of Coupling from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_y\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_coupling.png">png</a>, <a href="./figs/opt_stiffness_wz_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -462,12 +469,12 @@ Note that we can use very soft nano-hexapod if we limit the spindle rotating spe
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'compliance'</span>);
|
||||
<pre class="src src-matlab"> initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'compliance'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -475,23 +482,23 @@ initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-
|
||||
We put nothing on top of the micro-hexapod.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
<pre class="src src-matlab"> initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
|
||||
The diagonal element of the identified Micro-Station compliance matrix are shown in Figure <a href="#org15a14d9">5</a>.
|
||||
The diagonal element of the identified Micro-Station compliance matrix are shown in Figure <a href="#org15a14d9">6</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org15a14d9" class="figure">
|
||||
<p><img src="figs/opt_stiff_micro_station_compliance.png" alt="opt_stiff_micro_station_compliance.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Identified Compliance of the Micro-Station (<a href="./figs/opt_stiff_micro_station_compliance.png">png</a>, <a href="./figs/opt_stiff_micro_station_compliance.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Identified Compliance of the Micro-Station (<a href="./figs/opt_stiff_micro_station_compliance.png">png</a>, <a href="./figs/opt_stiff_micro_station_compliance.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -505,7 +512,7 @@ This is equivalent of identifying the dynamics of the nano-hexapod when fixed to
|
||||
We also choose the sample to be rigid and to have a mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 10);
|
||||
<pre class="src src-matlab"> initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -513,7 +520,7 @@ We also choose the sample to be rigid and to have a mass of 10kg.
|
||||
As before, we identify the dynamics for the following actuator stiffnesses:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
<pre class="src src-matlab"> Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -536,38 +543,38 @@ We plot the change of dynamics due to the compliance of the Micro-Station.
|
||||
The solid curves are corresponding to the nano-hexapod without the micro-station, and the dashed curves with the micro-station:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org6257db9">6</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#orgcea6354">7</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#org5c6f89c">8</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org7f2360c">9</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org6257db9">7</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#orgcea6354">8</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#org5c6f89c">9</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org7f2360c">10</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org6257db9" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_iff.png" alt="opt_stiffness_micro_station_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_micro_station_iff.png">png</a>, <a href="./figs/opt_stiffness_micro_station_iff.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_iff.png">png</a>, <a href="./figs/opt_stiffness_micro_station_iff.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgcea6354" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_dvf.png" alt="opt_stiffness_micro_station_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_micro_station_dvf.png">png</a>, <a href="./figs/opt_stiffness_micro_station_dvf.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_dvf.png">png</a>, <a href="./figs/opt_stiffness_micro_station_dvf.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5c6f89c" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_fx_dx.png" alt="opt_stiffness_micro_station_fx_dx.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_micro_station_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fx_dx.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fx_dx.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7f2360c" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_fz_dz.png" alt="opt_stiffness_micro_station_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_micro_station_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fz_dz.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -591,15 +598,15 @@ When the nano-hexapod is stiff (\(k>10^7\ [N/m]\)), the compliance of the micro-
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org37d23dd" class="outline-3">
|
||||
<h3 id="org37d23dd"><span class="section-number-3">3.1</span> Initialization</h3>
|
||||
<div id="outline-container-orgc1416f3" class="outline-3">
|
||||
<h3 id="orgc1416f3"><span class="section-number-3">3.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
We don’t include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. :exports none
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
<pre class="src src-matlab"> initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -607,10 +614,10 @@ We don’t include disturbances in this model as it adds complexity to the s
|
||||
We set the controller type to Open-Loop, and we do not need to log any signal.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeController();
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeReferences();
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeController();
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeReferences();
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -632,8 +639,8 @@ We make the following change of payload dynamics:
|
||||
We identify the dynamics for the following payload masses <code>Ms</code> and nano-hexapod leg’s stiffnesses <code>Ks</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ms = [1, 20, 50]; <span class="org-comment">% [Kg]</span>
|
||||
Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
<pre class="src src-matlab"> Ms = [1, 20, 50]; <span class="org-comment">% [Kg]</span>
|
||||
Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -641,7 +648,7 @@ Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
We then identify the dynamics for the following payload resonance frequencies <code>Fs</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fs = [50, 200, 500]; <span class="org-comment">% [Hz]</span>
|
||||
<pre class="src src-matlab"> Fs = [50, 200, 500]; <span class="org-comment">% [Hz]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@ -658,12 +665,12 @@ We then identify the dynamics for the following payload resonance frequencies <c
|
||||
We here compare the dynamics for the same payload mass, but different stiffness resulting in different resonance frequency of the payload:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orgcce8740">10</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org87ecfc8">11</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
<li>Figure <a href="#orgcce8740">11</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org87ecfc8">12</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740">10</a>):
|
||||
We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740">11</a>):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>The first mass line corresponds to \(\frac{1}{(m_n + m_p)s^2}\) where \(m_p = 10\ [kg]\) is the mass of the payload and \(m_n = 15\ [Kg]\) is the mass of the nano-hexapod top platform and attached mirror</li>
|
||||
@ -675,14 +682,14 @@ We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740
|
||||
<div id="orgcce8740" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_freq_fz_dz.png" alt="opt_stiffness_payload_freq_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_freq_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_fz_dz.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_freq_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org87ecfc8" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_freq_all.png" alt="opt_stiffness_payload_freq_all.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency (<a href="./figs/opt_stiffness_payload_freq_all.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_all.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency (<a href="./figs/opt_stiffness_payload_freq_all.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_all.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -694,8 +701,8 @@ We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740
|
||||
We here compare the dynamics for different payload mass with the same resonance frequency (100Hz):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org4eb6bfc">12</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org6e013a7">13</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
<li>Figure <a href="#org4eb6bfc">13</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org6e013a7">14</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
@ -712,14 +719,14 @@ We can see here that for the soft nano-hexapod:
|
||||
<div id="org4eb6bfc" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_mass_fz_dz.png" alt="opt_stiffness_payload_mass_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_mass_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_fz_dz.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_mass_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6e013a7" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_mass_all.png" alt="opt_stiffness_payload_mass_all.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass (<a href="./figs/opt_stiffness_payload_mass_all.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_all.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass (<a href="./figs/opt_stiffness_payload_mass_all.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_all.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -728,7 +735,7 @@ We can see here that for the soft nano-hexapod:
|
||||
<h4 id="org632af10"><span class="section-number-4">3.3.3</span> Total variation</h4>
|
||||
<div class="outline-text-4" id="text-3-3-3">
|
||||
<p>
|
||||
We now plot the total change of dynamics due to change of the payload (Figure <a href="#org75c1705">14</a>):
|
||||
We now plot the total change of dynamics due to change of the payload (Figures <a href="#orgf33beff">15</a> and <a href="#org75c1705">16</a>):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>mass from 1kg to 50kg</li>
|
||||
@ -736,10 +743,17 @@ We now plot the total change of dynamics due to change of the payload (Figure <a
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgf33beff" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_impedance_all_fz_dz.png" alt="opt_stiffness_payload_impedance_all_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_impedance_all_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_impedance_all_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org75c1705" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_impedance_fz_dz.png" alt="opt_stiffness_payload_impedance_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_impedance_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_impedance_fz_dz.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_impedance_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_impedance_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -771,63 +785,63 @@ We now consider the total change of nano-hexapod dynamics due to:
|
||||
The obtained dynamics are shown:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org045955d">15</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
|
||||
<li>Figure <a href="#orgc2c0741">16</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org7a8f7c3">17</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org4745a60">18</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org045955d">17</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
|
||||
<li>Figure <a href="#orgc2c0741">18</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org7a8f7c3">19</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org4745a60">20</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
And finally, in Figures <a href="#org0a4e875">19</a> and <a href="#orge890242">20</a> are shown an animation of the change of dynamics with the nano-hexapod’s stiffness.
|
||||
And finally, in Figures <a href="#org0a4e875">21</a> and <a href="#orge890242">22</a> are shown an animation of the change of dynamics with the nano-hexapod’s stiffness.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org045955d" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e3.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^3\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^3\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc2c0741" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e5.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^5\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^5\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7a8f7c3" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e7.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^7\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^7\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org4745a60" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e9.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^9\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^9\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org0a4e875" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_task_space.gif" alt="opt_stiffness_plant_dynamics_task_space.gif" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge890242" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_task_space_colors.gif" alt="opt_stiffness_plant_dynamics_task_space_colors.gif" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-03 ven. 17:55</p>
|
||||
<p class="date">Created: 2020-04-05 dim. 19:43</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
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Reference in New Issue
Block a user